Train Routing Problem Research Articles

Extensive Model and Matheuristic Algorithm for the Train Platforming Problem with Two-Train-Capacity Tracks: A Case Study of Prague Central Station

This paper provides a deeper insight into the train platforming problem (TPP). Many studies have focused on different versions of train scheduling and routing problems, and most of them assume that the platform track’s capacity is one train. However, especially in busy and complex railway stations, most platform tracks are divided into two parts, allowing two trains to simultaneously share the same platform track for passenger boarding/alighting. This results in more efficient train assignment to the platform tracks. In addition, consideration of the track capacity makes the problem more difficult because directions of trains are problematic. Motivated by this challenge, we consider the TPP with two-train-capacity tracks. We first describe the problem in detail and then propose a mixed-integer programming model. The objective of the considered problem is to minimize the total weighted train delays, which are defined as the difference between the departure times calculated by the mathematical model (M1) and the scheduled departure times of the trains in the timetable. Because of the NP-hard nature of the problem, the proposed M1 may not find feasible solutions for large-size problems. Thus, a matheuristic algorithm (MA) is developed to solve large-size problems. We used randomly generated test problems to demonstrate the performance of the proposed M1 and MA. Experimental results showed that MA outperforms M1 in both solution quality and solution time. Additionally, a case study was conducted at the central station of Prague, Czechia.

Battery energy storage train routing and security constrained unit commitment under solar uncertainty

The location of solar parks far from load areas may lead to transmission congestion and thus solar curtailment for secure system operation. Battery energy storage (BES) Train as mobile storage can transmit solar energy from site to load centers using a transport network while relieving lines from congestion. Therefore, stochastic security-constrained unit commitment (SCUC) with BES Train is modeled in this paper under solar uncertainty and N-1 critical line contingency. Time-space network models transportation constraints of BES Train routing problem. To reduce the computational burden, this paper proposes modeling of line outage distribution factor (LODF) for DC power flow that reduces the nonzero coefficients of post-contingency network constraints. The benders decomposition technique simplifies stochastic optimization for the mixed-integer linear programming (MILP) problem. BES Train integrated modified IEEE reliability test and 118-bus system is investigated as a case study. Simulations evaluate the effect of BES Train, solar uncertainty and critical line contingency state with LODF on operation costs, BES Train scheduling, solar curtailment, congestion and computational time. The proposed model shows that BES Train is an economical option for large solar integration to reduce transmission congestion and curtail solar power effectively.

Integrated train routing and timetabling problem in a multi-station high-speed railway hub

ABSTRACT With multiple stations being built in the high-speed railway hub, how to schedule trains among stations is of great significance. This paper studies the integrated train routing and timetabling problem in a multi-station high-speed railway hub. Based on a specific bi-level network, a binary integer programming model using cumulative flow variables is proposed to minimize the total cost of trains and passengers. An extended Alternating Direction Method of Multipliers (ADMM) algorithm is developed to decompose the model into train and passenger subproblems. The performance of the algorithm is investigated on small-scale and large-scale networks. Compared with the commonly used Lagrangian Relaxation (LR) algorithm, the results indicate that the extended ADMM obtain better solutions with less iterations, where the lower bound decreases by 3.79% and iterations are reduced by 83.54%. Another comparative experiment shows the total cost of the optimization scheme is 16.22% lower than that of the current fixed scheme.

Train Routing and Track Allocation Optimization Model of Multi-Station High-Speed Railway Hub

A multi-station high-speed railway hub is the interaction of multiple high-speed railway lines, and its operation determines the efficiency of the whole network. In this study, we focus on the train routing problem for a multi-station railway hub and propose a flexible scheme to improve the capacity utilization. Based on the flexible scheme, a mixed-integer programming node-arc model is formulated to minimize the total cost of train and passenger routes. Specifically, for diversifying the train routes, we consider the individual train rather than the train flow as the basic unit, which differs from the approach in previous studies. For each train route, in addition to the macro-scale route between stations, the micro-scale track allocation inside stations is also considered. Afterward, the optimization solver Gurobi is used to solve the model and obtain the optimization scheme. A case study based on real data from the Zhengzhou railway hub in China is implemented to evaluate the effectiveness of the optimization model. By comparing the optimization scheme with the fixed scheme and the sequential scheme, the results show our method could reduce the total cost by 29.35% and 22.58%, and the line and track capacity utilization of the optimization scheme is more reasonable. We provide some suggestions to help railway managers improve the operation efficiency and service quality of multi-station high-speed railway hubs.

Optimality analysis of train platforming and routing with different interlocking modes

The train platforming and routing (TPR) problem is to decide train operations within stations after the network-wise train schedules are determined. A feasible TPR plan requires both platform and route conflict-free, where the avoidance of route conflict is controlled according to three interlocking modes. Although the TPR problem is widely studied, none of them did a serious investigation on the optimality impacts of different interlocking modes in the TPR. Therefore, this paper introduced and formulated a space–time version of TPR considering three interlocking modes and subsequentially conducted numerical experiments to analyze the optimality differences under each mode. Based on the experimental findings, engineering practical suggestions are also provided. In summary, the experiment results showed that both route-locking sectional-release and sectional-locking sectional-release modes significantly outperform the route-locking route-release mode. And among them, using the route-locking sectional-release mode can bring notable benefits on large stations with high-density volumes while using the sectional-locking sectional-release mode can always provide outstanding outcomes over various station and traffic settings.

Train Operations Organization in High-Speed Railway Station Considering Variable Configuration

The train operation problem at the railway station is one of the basic scheduling problems for railway companies, and the decomposition and combination of trains in station platform tracks is an important trend in the railway operation, which further increase the complexity of train routing problem in the high-speed railway station. A reasonable train routing plan is critical for ensuring the safety and efficiency of train operations within the stations. Considering the combination and decomposition process of trains at the station, a multi-objective model is constructed, which determines the receiving routes and departure routes of trains at the railway station with the limited infrastructure capacity, given the station yard layouts, train diagram and relevant technical operation time of the station. A solution framework based on the “constraint method” is designed, and software Gurobi is used to solve the problem. The Pareto optimal solution set is obtained. The results show that when making 1.9% concessions to the minimum route cost, the occupancy balance of platform tracks can be improved by 66.2%, which is taken as the comprehensive application scheme. Compared with the feasible scheme, the comprehensive operation scheme reduces the route cost by 1.9% and improves the occupation balance of platform tracks by 15.4%.

Optimizing Train Routing Problem in a Multistation High-Speed Railway Hub by a Lagrangian Relaxation Approach

As the intersection of multiple high-speed railway lines, the multi-station high-speed railway hub is the key to improve the transport efficiency of the high-speed railway network. This paper focuses on the optimization of the multi-station high-speed railway hub and models it as a train routing problem (TRP). Considering the capacity of railway infrastructures and the demand of passengers, a mixed integer linear programming model is proposed to minimize the total cost of train routes and passenger routes. The optimized train routes include the macroscopic routes between stations and the microscopic track allocation inside stations and Electric Multiple Units (EMUs) depots. A Lagrangian relaxation (LR) approach is developed to dualize the hard constraints and decompose the origin model into train and passenger subproblems, then a shortest path algorithm is designed to solve the subproblems independently. Numerical experiments based on an illustrative railway hub network and a real-world network are implemented to demonstrate the effectiveness of the model and algorithm. The solution results prove that the LR approach can obtain high-quality solutions within an acceptable computational time. Compared with the existing fixed scheme, the optimization scheme can reduce the total cost by 37.18% and utilize the railway lines and tracks more reasonably.

Optimizing train frequencies and train routing with simultaneous passenger assignment in high-speed railway network

In a high speed railway line plan (LP), train frequencies, train routing and passenger assignment problems are usually solved sequentially, which may lower the quality of the line plan. To bridge the gap, a bi-level multi-objective mixed integer nonlinear programming model for simultaneously obtaining the optimal plan of passenger assignment and train routing is developed in this paper. The upper level model is an integer programming (IP) model with an objective of minimizing both the total train travel time and the average capacity utilization rate, while the lower level one is a model with an objective of minimizing the total travel time of passengers and maximizing the average train attendance rate for all sections. In order to obtain the solution in relatively short time, a particle swarm approach is adopted to solve the model in a large scale railway network. As a case study, the model is applied to a high speed railway network in China. The results show that the total travel time of passengers and trains obtained from the simultaneous optimization model can be reduced by 0.9% in comparison with the two problems solved sequentially, which indicates that the model and the solution algorithms could be useful in practice.

Two-Stage Robust-Stochastic Electricity Market Clearing Considering Mobile Energy Storage in Rail Transportation

This paper proposes a two-stage robust-stochastic framework to evaluate the effect of the battery-based energy storage transport (BEST) system in a day-ahead market-clearing model. The model integrates the energy market-clearing process with a train routing problem, where a time-space network is used to describe the limitations of the rail transport network (RTN). Likewise, a price-sensitive shiftable (PSS) demand bidding approach is applied to increase the flexibility of the power grid operation and reduce carbon emissions in the system. The main objective of the proposed model is to determine the optimal hourly location, charge/discharge scheduling of the BEST system, power dispatch of thermal units, flexible loads scheduling as well as finding the locational marginal price (LMP) considering the daily carbon emission limit of thermal units. The proposed two-stage framework allows the market operator to differentiate between the risk level of all existing uncertainties and achieve a more flexible decision-making model. The operator can modify the conservatism degree of the market-clearing using a non-probabilistic method based on info-gap decision theory (IGDT), to reduce the effect of wind power fluctuations in real-time. In contrast, a risk-neutral-based stochastic technique is used to meet power demand uncertainty. The results of the proposed mixed-integer linear programming (MILP) problem, confirm the potential of BEST and PSS demand in decreasing the LMP, line congestion, carbon emission, and daily operation cost.

A Multiobjective Optimization for Train Routing at the High-Speed Railway Station Based on Tabu Search Algorithm

This paper focuses on the train routing problem at a high-speed railway station to improve the railway station capacity and operational efficiency. We first describe a node-based railway network by defining the turnout node and the arrival-departure line node for the mathematical formulation. Both considering potential collisions of trains and convenience for passengers’ transfer in the station, the train routing problem at a high-speed railway station is formulated as a multiobjective mixed integer nonlinear programming model, which aims to minimize trains’ departure time deviations and total occupation time of all tracks and keep the most balanced utilization of arrival-departure lines. Since massive decision variables for the large-scale real-life train routing problem exist, a fast heuristic algorithm is proposed based on the tabu search to solve it. Two sets of numerical experiments are implemented to demonstrate the rationality and effectiveness of proposed method: the small-scale case confirms the accuracy of the algorithm; the resulting heuristic proved able to obtain excellent solution quality within 254 seconds of computing time on a standard personal computer for the large-scale station involving up to 17 arrival-departure lines and 46 trains.

Multi-objective mathematical modeling of an integrated train makeup and routing problem in an Iranian railway company

Train formation planning faces two types of challenges; namely, the determination of the quantity of cargo trains run known as the frequency of cargo trains and the formation of desired allocations of demands to a freight train. To investigate the issues of train makeup and train routing simultaneously, this multi-objective model optimizes the total profit, satisfaction level of customers, yard activities in terms of the total size of a shunting operation, and underutilized train capacity. It also considers the guarantee for the yard-demand balance of flow, maximum and minimum limitations for the length of trains, maximum yard limitation for train formation, maximum yard limitation for operations related to shunting, maximum limitation for the train capacity, and upper limit of the capacity of each arc in passing trains. In this paper, a goal programming approach and an L­p ­ norm method are applied to the problem. Furthermore, a simulated annealing (SA) algorithm is designed. Some test problems are also carried out via simulation and solved using the SA algorithm. Furthermore, a sample investigation is carried out in a railway company in Iran. The findings show the capability and performance of the proposed approach to solve the problems in a real rail network.

Train routing and timetabling problem for heterogeneous train traffic with switchable scheduling rules

This paper proposes a mathematical model for the train routing and timetabling problem that allows a train to occasionally switch to the opposite track when it is not occupied, which we define it as switchable scheduling rule. The layouts of stations are taken into account in the proposed mathematical model to avoid head-on and rear-end collisions in stations. In this paper, train timetable could be scheduled by three different scheduling rules, i.e., no switchable scheduling rule (No-SSR) which allows trains switching track neither at stations and segments, incomplete switchable scheduling rule (In-SSR) which allows trains switching track at stations but not at segments, and complete switchable scheduling rule (Co-SSR) which allows trains switching track both at stations and segments. Numerical experiments are carried out on a small-scale railway corridor and a large-scale railway corridor based on Beijing–Shanghai high-speed railway (HSR) corridor respectively. The results of case studies indicate that Co-SSR outperforms the other two scheduling rules. It is also found that the proposed model can improve train operational efficiency.

Algorithms for a special class of state-dependent shortest path problems with an application to the train routing problem

We study the state-dependent shortest path problem and focus on its application to the Single Train Routing Problem consisting of a rail network with only double-track segments, where the objective is to route one train through an empty network as fast as possible. We show that the Single Train Routing Problem is NP-hard. We investigate the solution properties and present sufficient conditions for optimality. Different conditions on the parameters are given to guarantee that certain local route selection is optimal. Then, a dynamic programming heuristic is introduced and conditions when the proposed heuristic can obtain the optimal solution in polynomial time are also discussed. Experimental results show the efficiency of the proposed heuristics for general problem settings.

Simultaneous passenger train routing and timetabling using an efficient train-based Lagrangian relaxation decomposition

This paper focuses on the simultaneous passenger train routing and timetabling problem on the rail network consisting of both unidirectional and bidirectional tracks using an efficient train-based Lagrangian relaxation decomposition. We first build an integer linear programming model with many 0–1 binary and non-negative integer decision variables, after then reformulate it as a train path-choice model for providing an easier train-based Lagrangian relaxation decomposition mechanism based on the construction of space-time discretized network extending from node-cell-based rail network. Moreover, through reformulating safety usage interval restrictions with a smaller number of constraints in this reformulated model, the train-based decomposition needs fewer Lagrangian multipliers to relax these constraints. On the basis of this decomposition, a solving framework including a heuristic algorithm is proposed to simultaneously optimize both the dual and feasible solutions. A set of numerical experiments demonstrate the proposed Lagrangian relaxation decomposition approach has better performances in terms of minimizing both train travel time and computational times.

Part supply method for mixed-model assembly lines with decentralized supermarkets

In-house part supply affects the efficiency of mixed-model assembly lines considerably. Hence, we propose a reliable Just-In-Time part supply strategy with the use of decentralized supermarkets. For a given production sequence and line layout, the proposed strategy schedules tow train routing and delivery problems jointly to minimize the number of employed town trains and the traveling time, while ensuring that stations never run out of parts. To solve this problem, a mathematical formulation is proposed for each sub-problem aiming at minimizing supply cost. Then, a dynamic programming algorithm for routing and a greedy algorithm for delivery are developed, both of which are of polynomial runtime. Finally, a computational study is implemented to validate the effectiveness of the strategy, and to investigate the effects of the delivery capacity of tow trains and storage capacity of stations on supply cost.

The Freight Train Routing Problem for Congested Railway Networks with Mixed Traffic

We consider the following freight train routing problem (FTRP). Given is a transportation network with fixed routes for passenger trains and a set of freight trains (requests), each defined by an origin and destination station pair. The objective is to calculate a feasible route for each freight train such that the sum of all expected delays and all running times is minimal. Previous research concentrated on microscopic train routings for junctions or inside major stations. Only recently approaches were developed to tackle larger corridors or even networks. We investigate the routing problem from a strategic perspective, calculating the routes in a macroscopic transportation network of Deutsche Bahn AG. In this context, macroscopic refers to an aggregation of complex and large real-world structures into fewer network elements. Moreover, the departure and arrival times of freight trains are approximated. The problem has a strategic character since it asks only for a coarse routing through the network without the precise timings. We provide a mixed-integer nonlinear programming (MINLP) formulation for the FTRP, which is a multicommodity flow model on a time-expanded graph with additional routing constraints. The model’s nonlinearities originate from an algebraic approximation of the delays of the trains on the arcs of the network by capacity restraint functions. The MINLP is reduced to a mixed-integer linear model (MILP) by piecewise linear approximation. The latter is solved by a state-of-the art MILP solver for various real-world test instances.

Lower and upper bound algorithms for the real-time train scheduling and routing problem in a railway network

Abstract: This paper focuses on the development of new algorithms for the real-time train scheduling and routing problem in a complex and busy railway network. Since this is a strongly NP-hard problem and practical size instances are complex, simple heuristics are typically adopted in practice to compute feasible but low quality schedules in a short computation time. In order to compute good quality solutions, we consider a mixed-integer linear programming formulation of the problem and solve it with a commercial solver. However, the resolution of this formulation by a commercial solver often takes a too long computation time. Therefore, a new methodology based on the relaxation of some train routing constraints in the formulation is proposed for the quick computation of a good quality lower bound. The lower bound solution is then transformed via a constructive metaheuristic into a feasible schedule, representing a good quality upper bound to the problem. Computational experiments are performed on several disturbed traffic situations for two practical case studies from the Dutch and British railways. The results show that the new lower and upper bounds are computed in a few seconds and are often of similar quality to the ones computed by the commercial solver in hours of computation.

Improving robustness of rolling stock circulations in rapid transit networks

The rolling stock circulation depends on two different problems: the rolling stock assignment and the train routing problems, which up to now have been solved sequentially. We propose a new approach to obtain better and more robust circulations of the rolling stock train units, solving the rolling stock assignment while accounting for the train routing problem. Here robustness means that difficult shunting operations are selectively penalized and propagated delays together with the need for human resources are minimized. This new integrated approach provides a huge model. Then, we solve the integrated model using Benders decomposition, where the main decision is the rolling stock assignment and the train routing is in the second level. For computational reasons we propose a heuristic based on Benders decomposition. Computational experiments show how the current solution operated by RENFE (the main Spanish train operator) can be improved: more robust and efficient solutions are obtained.

Multi-objective optimization of train routing problem combined with train scheduling on a high-speed railway network

Based on train scheduling, this paper puts forward a multi-objective optimization model for train routing on high-speed railway network, which can offer an important reference for train plan to provide a better service. The model does not only consider the average travel time of trains, but also take the energy consumption and the user satisfaction into account. Based on this model, an improved GA is designed to solve the train routing problem. The simulation results demonstrate that the accurate algorithm is suitable for a small-scale network, while the improved genetic algorithm based on train control (GATC) applies to a large-scale network. Finally, a sensitivity analysis of the parameters is performed to obtain the ideal parameters; a perturbation analysis shows that the proposed method can quickly handle the train disturbance.

High-Speed Train Network Routing with Column Generation

This paper is based on an award-winning rail scheduling project. It examines a high-speed train routing problem in which the trains are operated on a large network with sparse stations. The relatively long distance between train stations makes it unlikely to get a backup train unit in time when an operating train is out of service because of the need for maintenance. Therefore, regular maintenance is carefully incorporated into the base routing plan and is strictly enforced. An integrated path-based routing model is proposed here to design train routes and scheduling optimally. The objective is to cover the published trains and timetable with the least cost without violating operational rules and maintenance requirements. The scheduling for each operational cycle needs to be completed in a timely manner and be implementable in practice. A heuristic is proposed to solve the problem in a column generation framework. Numerical tests are conducted with operational data from the French National Railways. The test not only shows a significant cost savings but also indicates its efficiency in real-world applications.